Chapter 3 Preferences

1. Chapter 3 Preferences Course: Microeconomics Text: Varian’s Intermediate Microeconomics

2. Introduction • Last chapter we talk about what is affordable or feasible to consumers. • This time we talk about preferences: what the consumer like more and what they like less. • As a rational agent, a consumer chooses the option in the budget set that is highest in their preference order (i.e. one likes the most).

3. f f ~ ~ Preference Relations • Let x, y are consumption bundles. • denotes strict preference so x y means that bundle x is preferred strictly to bundle y. • ~ denotes indifference; x ~ y means x and y are equally preferred. • denotes weak preference;x y means x is preferred at least as much as is y. p p

4. Preference Relations • Strict preference, weak preference and indifference are all preference relations. • Particularly, they are ordinal relations; i.e. they state only the order in which bundles are preferred. • It has no indication of how much they like one versus the other.

5. f f f f ~ ~ ~ ~ Preference Relations • x y and y x imply x ~ y. • x y and (not y x) imply x y. p

6. f f ~ ~ Assumptions about Preference Relations • Completeness: For any two bundles x and y it is always possible to make the statement that either x y or y x. • Bundles are always comparable. • If both are true, then they are indifferent to the individual.

7. f ~ Assumptions about Preference Relations • Reflexivity: Any bundle x is always at least as preferred as itself; i.e. x x.

8. f f f ~ ~ ~ Assumptions about Preference Relations • Transitivity: Ifx is at least as preferred as y, andy is at least as preferred as z, thenx is at least as preferred as z; i.e.x y and y z x z. • It avoids circular preference, and ensure that there exists the best bundle.

9. f ~ Indifference Curves • Take a reference bundle x’. The set of all bundles equally preferred to x’ is the indifference curve (set) containing x’; the set of all bundles {y: y ~ x’}. • Weakly preferred set: bundles that are weakly preferred to x’. {y: y x’}.

10. Indifference Curves x2 x’ ~ x” ~ x”’ x’ The consumer is indifferent between every point on the indifference curve. x” x”’ x1

11. Indifference Curves zxy p p If consumer prefers more to less for each goods, all bundles on the northeast of the indifference curve are strictly preferred to x, and all bundles southwest of the indifference curve are less preferred to x. x2 x z y x1

12. Indifference Curves I1 All bundles in I1 are strictly preferred to all in I2. x2 x z I2 All bundles in I2 are strictly preferred to all in I3. y I3 x1

13. Indifference Curves x2 WP(x), the set of bundles weakly preferred to x. x I(x’) I(x) x1

14. Indifference Curves x2 WP(x), the set of bundles weakly preferred to x. x WP(x) includes I(x). I(x) x1

15. Indifference Curves x2 SP(x), the set of bundles strictly preferred to x, does not include I(x). x I(x) x1

16. Indifference Curves Cannot Intersect From I1, x ~ y. From I2, x ~ z. Therefore y ~ z. But because I1 and I2represent distinct level of preference, we seey z, a contradiction. I2 x2 I1 p x y z x1

17. Goods • When more of a commodity is always preferred, the commodity is a good. • If every commodity is a good then indifference curves are negatively sloped. • It is because when one has more of one good, one has to get less of another to make this bundle indifferent to the original one.

18. Slopes of Indifference Curves Good 2 Two goodsa negatively sloped indifference curve. Better Worse Good 1

19. Bads • If less of a commodity is always preferred then the commodity is a bad. • e.g. rotten fruits; tobacco smoke (if you do not smoke) • If one good is good and the other is bad, then the indifference curve would be upward sloping.

20. Slopes of Indifference Curves Good 2 One good and onebad a positively sloped indifference curve. Better Worse Bad 1 If you want more of the good, you also have to get more of the bad so that you are indifferent between them.

21. Neutrals • If one just do not care about whether or how much to have a commodity, this is called a neutral good. • E.g.: goods that you don’t use and do not care about their existence. • If one commodity is neutral, the other is good, the indifference curve would be vertical / horizontal.

22. Slope of Indifference Curve

23. Extreme Cases of Indifference Curves; Perfect Substitutes • If a consumer always regards units of commodities 1 and 2 as equivalent, then the commodities are perfect substitutes. • Only the total amount (or a weighted sum) of the two commodities in bundles determines their preference rank-order. • Example: orange juice of two different brands. Apartment in different locations.

24. Extreme Cases of Indifference Curves; Perfect Substitutes x2 Slopes are constant at - 1. 15 • I2 Bundles in I2 all have a totalof 15 units and are strictly preferred to all bundles in I1, which have a total of only 8 units in them. 8 I1 x1 8 15

25. Extreme Cases of Indifference Curves; Perfect Complements • If a consumer always consumes commodities 1 and 2 in fixed proportion (e.g. one-to-one), then the commodities are perfect complements. • Only the number of pairs of units of the two commodities determines the preference rank-order of bundles. • E.g.: left shoes/right shoes; computer and monitor.

26. Extreme Cases of Indifference Curves; Perfect Complements x2 Each of (5,5), (5,9) and (9,5) contains5 pairs so each is equally preferred. 45o 9 5 I1 x1 5 9

27. Extreme Cases of Indifference Curves; Perfect Complements x2 Since each of (5,5), (5,9) and (9,5) contains 5 pairs, each is less preferred than the bundle (9,9)which contains 9 pairs. 45o 9 I2 5 I1 x1 5 9

28. Preferences Exhibiting Satiation • A bundle strictly preferred to any others is a satiation point or a bliss point. • The satiation point is the best bundle. More of anything are not better. • What do indifference curves look like for preferences exhibiting satiation?

29. Indifference Curves Exhibiting Satiation x2 Satiation(bliss)point x1

30. Indifference Curves Exhibiting Satiation x2 Better Better Satiation(bliss)point Better x1

31. Indifference Curves Exhibiting Satiation x2 Better Better Satiation(bliss)point Better x1

32. Indifference Curves for Discrete Commodities • A commodity is infinitely divisible if it can be acquired in any quantity; e.g. water or cheese. • A commodity is discrete if it comes in unit lumps of 1, 2, 3, … and so on; e.g. aircraft, ships and refrigerators.

33. Indifference Curves for Discrete Commodities • Suppose commodity 2 is an infinitely divisible good (gasoline) while commodity 1 is a discrete good (aircraft). What do indifference “curves” look like?

34. Indifference Curves With a Discrete Good Gasoline Indifference “curves”are collections ofdiscrete points. Aircraft 0 1 2 3 4

35. Well-Behaved Preferences • Typical assumptions of preferences • A preference relation is “well-behaved” if it is • monotonic and convex. • Monotonicity: More of any commodity is always preferred (i.e. no satiation and every commodity is a good). • This implies a negatively sloped IC.

36. Well-Behaved Preferences • Convexity: Mixtures of bundles are (at least weakly) preferred to the bundles themselves. E.g., the 50-50 mixture of the bundles x and y is z = (0.5)x + (0.5)y.z is at least as preferred as x or y.

37. Well-Behaved Preferences -- Convexity. x x2 x+y Is (weakly) preferred to both x and y. x2+y2 z = 2 2 y y2 x1+y1 x1 y1 2

38. Well-Behaved Preferences -- Convexity. x x2 z =(tx1+(1-t)y1, tx2+(1-t)y2) is preferred to x and y for all 0 < t < 1. y y2 x1 y1

39. Well-Behaved Preferences -- Convexity. Preferences are strictly convex when all mixtures z are strictly preferred to their component bundles x and y. x x2 z y y2 x1 y1

40. Non-Convex Preferences The mixture zis less preferred than x or y. One likes both, but does want to consume together. x2 Better z y2 x1 y1

41. More Non-Convex Preferences x2 Better The mixture zis less preferred than x or y. z y2 x1 y1

42. Why convexity? • It represents a more balanced preference rather than a preference that induces a specialization. • It is natural to consume the goods involved in positive amount. • It also implies a diminishing marginal rate of substitution.

43. Slopes of Indifference Curves • The negative of the slope of an indifference curve is its marginal rate of substitution (MRS). • Note: this is slightly different from the textbook, but my definition is more popular. • This represents the maximum amount of x2 one is willing to give up per unit of x1 at a specific consumption bundle.

44. Marginal Rate of Substitution x2 MRS at x’ is the (negative) slope of theindifference curve at x’ x’ x1

45. Marginal Rate of Substitution x2 MRS at x’ is lim -{Dx2/Dx1}Dx1 0= -dx2/dx1 at x’ x’ Dx2 Dx1 x1

46. Marginal Rate of Substitution -dx2= MRS ´ dx1 so, at x’, MRS is the rate at which the consumer is only just willing to exchange commodity 2 for a small amount of commodity 1. x2 x’ dx2 dx1 x1

47. MRS & Ind. Curve Properties Good 2 Two goodsa negatively sloped indifference curve Better MRS > 0. Worse Good 1

48. MRS & Ind. Curve Properties Good 2 One good and onebad a positively sloped indifference curve Better MRS < 0. Worse Bad 1 Because instead of giving up, you have to obtain more good 2 for you to be willing to accept more good 1.

49. MRS & Ind. Curve Properties Good 2 MRS = 5 MRS always decreases with x1if and only if preferences are strictlyconvex. MRS = 0.5 Good 1 We call it a diminishing marginal rate of substitution.

50. MRS & Ind. Curve Properties x2 MRS increasesas x1 increasesnon-convex preferences MRS = 0.5 MRS = 5 x1